Prove that the circumcircle of a triangle contains exactly 3 points whose Simson lines are tangent to the triangle's Euler circle and these points are the vertices of an equilateral triangle.
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http://www.artofproblemsolving.com/Forum/viewtopic.php?t=25115
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=47&t=107065
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=47&t=244550
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=47&t=277289